Starlight Tools

Eigenvalue and Eigenvector Calculator with Steps

Find real or complex eigenvalues, eigenspace bases, and multiplicities for a 2×2 or 3×3 real matrix. Every calculation stays in your browser.

Enter a Square Matrix

Enter real integers, decimals, or scientific notation. Complex eigenpairs are supported even though matrix entries must be real.

Put rows on separate lines and separate entries with spaces, commas, or tabs. A valid 2×2 or 3×3 matrix changes the size automatically.

Eigenvalues and Eigenvectors

Characteristic polynomialdet(λI − A) = —

Enter a valid matrix to calculate its eigenpairs.

Results will appear here without changing the page layout.

Step-by-Step Working

Calculate a matrix to see its characteristic equation, root method, reduced systems, eigenspace bases, and residual checks.

Eigenvalue and Eigenvector Formulas

Characteristic equation

det(λI − A) = 0

Its roots are the eigenvalues. A 2×2 matrix gives a quadratic and a 3×3 matrix gives a cubic.

Eigenvectors

(A − λI)v = 0, v ≠ 0

For each eigenvalue, the null space of A − λI supplies one or more independent eigenvectors.

Verification

Av = λv

The calculator reports max|Av − λv|. A value near zero confirms the displayed eigenpair to working precision.

Numerical convention: Calculations use JavaScript double-precision arithmetic. Inputs are internally scaled before solving. Eigenvectors are normalized to Euclidean length 1 and their complex phase is chosen for a readable leading component. Very close or repeated roots and nearly defective matrices can be sensitive to rounding.

How to Use the Calculator

  1. Select 2×2 or 3×3 and enter every real matrix value, or paste a complete matrix.
  2. Choose the number of displayed decimal places; this does not round intermediate calculations.
  3. Select Calculate eigenpairs to solve the characteristic equation.
  4. Review each eigenvalue, normalized eigenspace basis, multiplicity, RREF system, and residual, then copy or download the working if needed.

Worked 2×2 Example

For A = [[4,1],[2,3]], the characteristic equation is:

det(λI − A) = (λ − 4)(λ − 3) − 2 = λ² − 7λ + 10 = 0

The roots are λ = 5 and λ = 2. Solving (A − 5I)v = 0 gives a direction proportional to [1,1]; solving (A − 2I)v = 0 gives one proportional to [1,−2].

Repeated roots

If an eigenvalue appears more than once, compare algebraic multiplicity with the number of independent eigenvectors. A shortfall means the matrix is defective.

Complex conjugate pairs

A real rotation matrix such as [[0,−1],[1,0]] has eigenvalues i and −i, with complex eigenvectors.

How to Interpret the Results

An eigenvector identifies a direction that a linear transformation does not turn away from itself in real or complex vector space. Its eigenvalue is the scale factor: magnitude greater than 1 expands, magnitude below 1 contracts, a negative real value reverses direction, and a complex value combines scaling with phase rotation.

Algebraic multiplicity is the number of times an eigenvalue occurs as a characteristic-polynomial root. Geometric multiplicity is the dimension of its eigenspace. A matrix is diagonalizable over the complex numbers when the total number of independent eigenvectors equals its size.

Eigenvalue and Eigenvector FAQs

What is an eigenvalue and eigenvector?

For a square matrix A, a nonzero vector v is an eigenvector if Av = λv. The scalar λ is its eigenvalue.

How does the calculator find eigenvalues?

It forms det(λI − A) and solves the resulting quadratic or cubic. It scales the input matrix first to reduce overflow and underflow risk, then scales the roots back.

Can a real matrix have complex eigenvalues?

Yes. Nonreal roots occur as complex-conjugate pairs for real matrices. The associated eigenvectors generally have complex components too.

What is the difference between algebraic and geometric multiplicity?

Algebraic multiplicity counts repeated characteristic roots. Geometric multiplicity counts independent eigenvectors for that root, or equivalently the dimension of null(A − λI).

Does every matrix have enough eigenvectors to diagonalize it?

No. A repeated eigenvalue can have fewer independent eigenvectors than its algebraic multiplicity. Such a matrix is defective and is not diagonalizable.

Why is the residual not exactly zero?

Decimal arithmetic, cubic roots, and row reduction use finite binary floating-point numbers. A small residual is expected; nearly repeated or ill-conditioned eigenpairs can have larger sensitivity.

Are my matrix values private?

Yes. Parsing, calculation, copying, and text-file creation occur locally in your browser. This page’s calculator code does not transmit or store matrix entries.

Calculation Notes and Limits

Last reviewed: August 5, 2026 by the Starlight Tools editorial team. Definitions and the det(λI − A) convention were checked against the NIST Digital Library of Mathematical Functions, §3.2(iv).

The calculator accepts finite real entries with magnitude no greater than 10⁹⁰ and matrices of size 2×2 or 3×3. It groups numerically indistinguishable roots using a relative tolerance and computes eigenspaces by complex row reduction. For exact symbolic proofs, high-precision research, or safety-critical engineering, verify results with appropriate specialist software.

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